Slope of a Line MCQ Quiz - Objective Question with Answer for Slope of a Line - Download Free PDF

Concept:

When two lines are perpendicular, then the product of their slope is -1.

i.e. m₁ × m₂ = -1

When the slope of a line is m then the slope of a line perpendicular to it = - 1/m

Calculation:

Given:

-2x + 3y + 4 = 0

y = \(\frac{2}{3}x-\frac{4}{3} \)

The slope of the line = \(\frac {2}{3}\)

The line perpendicular to the above line will have a slope of = \(-\frac {1}{\frac 23}\) = \(-\frac{3}{2}\)

option (1) 3x + 2y – 4 = 0 

• The slope of the line is -3/2

option (2) 3x – 2y + 4 = 0

• The slope of the line is 3/2

option (3)  -3x + 2y + 7 = 0

• The slope of the line is 3/2

option (4) 3x – 2y – 7 = 0

• The slope of the line is 3/2

Hence option (1) 3x + 2y – 4 = 0 is correct.

Explanation:

Slope of diagonal along the line x – 2y = 1

⇒m₁ = 1/2

Slope of the diagonal along the line 4x + 2y = 3

⇒m₂ = -2

Now,  m₁m₂ = \(\frac{1}{2}(-2) = -1\)

Then, Diagonals are perpendicular. 

∴ The quadrilateral ABCD is a rhombus.

Concept:

Angle Between Two Lines and Y-Intercept:

• The problem involves finding the y-intercept of a line L passing through a given point and making an angle of 60° with another line.
• The angle between two lines is given by the formula:
• tan(θ) = |(m₁- m₂) / (1 + m₁*m₂)|, where m₁ and m₂ are the slopes of the two lines.
• Once the equation of the line L is determined, the y-intercept can be found.

Calculation:

Given the line 2x - y + 3 = 0, we first calculate the slope of this line:

Rearranging the equation in slope-intercept form:

y = 2x + 3

The slope (m1) of this line is 2.

The line L passes through the point (2, 0) and makes an angle of 60° with the given line. The slope of line L, m2, can be calculated using the angle formula:

tan(60°) = |(m₂ - 2) / (1 + 2*m₂)|

Since tan(60°) = √3, the equation becomes:

√3 = |(m₂ - 2) / (1 + 2m₂)|

We now solve this equation for m₂:

√3(1 + 2m₂) = |m₂ - 2|

For simplicity, assume m₂ - 2 > 0 (since the angle is acute and the line makes an acute angle with the positive X-axis,   m2 > 2). Hence:

√3(1 + 2 m2) =  m2 - 2

√3 + 2√3* m2 =  m2 - 2

2√3* m2 -  m2 = -2 - √3

 m2(2√3 - 1) = -2 - √3

 m2 = (-2 - √3) / (2√3 - 1)

Now we calculate the y-intercept of the line L. The equation of the line L is:

y - 0 =  m2(x - 2)

y =  m2(x - 2)

Substitute x = 0 to find the y-intercept:

y =  m2(0 - 2)

y = -2 *  m2

Substitute the value of  m2 into this equation to calculate the y-intercept.

Hence, the y-intercept of the line L is:

The correct answer is 16 - 10√3 / 11

Concept Used:

1. A line with an undefined slope is a vertical line.

2. The slope of a line is given by m = tan θ, where θ is the angle made by the line with the positive X-axis.

3. The angle between two lines with slopes m₁ and m₂ is given by \(tan θ = |\frac{m_1 - m_2}{1 + m_1m_2}|\)

Calculation:

Given:

Line L passes through (-2, -3) and its slope is undefined.

Since L is vertical, its equation is x = -2.

The slope of the line ax - 2y + 3 = 0 is \(\frac{a}{2}\)

The angle between L and ax - 2y + 3 = 0 is 45°.

Since L is vertical, the line ax - 2y + 3 = 0 must be inclined at 45° or 135° to the y-axis.

⇒ \(\frac{a}{2}\) = tan 45° = 1 or \(\frac{a}{2}\) = tan 135° = -1

Since a > 0, \(\frac{a}{2}\) = 1 ⇒ a = 2

The line x + ay - 4 = 0 becomes x + 2y - 4 = 0.

The slope of this line is \(- \frac{1}{2}\)

Let θ be the angle made by this line with the positive X-axis.

⇒ tan θ = \(- \frac{1}{2}\)

Since the slope is negative, the angle is obtuse.

⇒ θ = \(\pi - tan^{-1}(\frac{1}{2})\)

∴ The angle made by the line x + ay - 4 = 0 with the positive X-axis is \(\pi - tan^{-1}(\frac{1}{2})\)

Hence option 1 is correct.

Answer : 1

Solution :

Understanding the properties of a line with a negative slope involves considering how slope affects the orientation of a line on a Cartesian plane. The slope of a line is defined as the ratio of the rise (vertical change) to the run (horizontal change), and it is typically represented as m in the slope-intercept equation of a line, y = mx + b.

A negative slope means that as one moves from left to right across the Cartesian plane, the line descends; it falls. This descent characteristic indicates that the line moves downward as it progresses horizontally. More formally, the angle & described here is the angle the line makes with the positive direction of the x-axis when moving in the clockwise direction.

Now, let's evaluate the given options:

• Option 1 : θ is an obtuse angle. Since an obtuse angle is greater than 90 degrees but less than 180 degrees, this represents the condition where a line has a negative slope, because the line descends as it moves from left to right, forming an obtuse angle with the x-axis. Hence, this option is applicable to lines with negative slopes.

• Option 2 : θ is equal to zero. This describes a line that lies coincident with the x-axis, i.e., the line has zero slope. Thus, a zero value of is not appropriate for negative slopes.

• Option 3 : Either the line is the x-axis or it is parallel to the x-axis. Lines that are the x-axis or parallel to it have a slope of zero, not negative. Therefore, this option does not describe lines with a negative slope.

• Option 4 : θ is an acute angle. An acute angle is less than 90 degrees, indicating a line with a positive slope if it forms such an angle with the x-axis. Thus, this option does not describe lines with a negative slope.

Therefore, the correct answer is Option 1 : θ is an obtuse angle, as this correctly describes the angle formed with the x-axis by a line that has a negative slope.

Concept:

• The slope of a line passing through the distinct points (x₁, y₁) and (x₂, y₂) is \(\frac{{{y_2}\; - \;{y_1}}}{{{x_2}\; - \;{x_1}}}\)
• When two lines are perpendicular, the product of their slope is -1. If m is the slope of a line, then the slope of a line perpendicular to it is -1/m.

Calculation:

Let the slope of the line 2x – 5y + 4 = 0 be m₁ and the slope of the line joining the points (1, 5) and (α, 3) be m₂ 

\( {{\rm{m}}_2} = \frac{{{\rm{3 }} - 5}}{{{\rm{α }} - 1}} = \frac{-2}{{{\rm{α }} - 1}}\)

Now, the slope of the line = m₁ = 2/5

Given lines are perpendicular to each other,

∴ m1 m2 = -1

\( ⇒ {\rm{\;}}\frac{{ - 2}}{{{\rm{α }} - 1}}{\rm{\;}} × {\rm{\;}}\frac{2}{5} = \; - 1\)

⇒ -4 = -5 × (α -1)

⇒ (α -1) = 4/5

⇒ α = (4/5) + 1 = 9/5

Concept:

The slope of the line joining the points (x1, y1) and (x2, y2) is given by: \(\rm m = \frac{{{y_2}\; - \;{y_1}}}{{{x_2} - {x_1}}}\)

Calculation:

Given: The slope of a line joining the points A (1, x) and B (3, 2) is 8

As we know, The slope of the line joining the points (x1, y1) and (x2, y2) is given by: \(\rm m = \frac{{{y_2}\; - \;{y_1}}}{{{x_2} - {x_1}}}\)

⇒ 8 = \(\rm \frac{{{2}\; - \;{x}}}{{{3} - {1}}}\)

⇒ 8 × 2 = 2 - x

⇒ 16 = 2 - x

∴ x = -14

Concept:

If θ is the inclination of the line l, then

The slope of a line is denoted by m = tan θ, θ ≠ 90°

Calculation:

Given: Line makes 30° with respect to the x-axis in a positive direction

∵ the inclination of the line is 30° i.e θ = 30° 

As we know that slope of a line is given by: m = tan θ 

Concept:

If two lines with slopes m and n are perpendicular to each other, then mn = -1.

The slope of a line passing through the points (x₁, y₁) and (x2, y2) is given by: m = \(\rm \frac{y_2-y_1}{x_2-x_1}\).

Calculation:

The slope of the line through the points (-1, 2) and (3, 6) is \(\rm \left(\frac{6-2}{3+1}\right)\)

The slope of the line through the points (4, 8) and (x, 12) is \(\rm \left(\frac{12-8}{x-4}\right)\)

Using the product of slopes of perpendicular lines, we get:

\(\rm \left(\frac{6-2}{3+1}\right)\left(\frac{12-8}{x-4}\right)=-1\)

⇒ 4 = (4 - x)

⇒ x = 0

Concept:

The general equation of a line is y = mx + c, where m is the slope. 

Calculation:

The given equation is 4x + 3y - 4 = 0

⇒ 3y = -4x + 4

⇒ y = (-4/3)x + 4/3

On compairing it with y = mx + c, we get

Slope, m = -4/3

Hence, the slope of the line is \(-\frac{4}{3}\).

CONCEPT:

The slope of a line is given by tan α and it is denoted by m.

i.e m = tan α, where α ≠ π/2 and it represents the angle which a given line makes with respect to the X – axis in the positive direction.

Note:

• Slope of X –axis or a line parallel to X – axis is : m = tan 0° = 0

• Slope of Y – axis or a line parallel to Y – axis is: m = tan π/2 = ∞

• The slope of the line joining the points (x1, y1) and (x2, y2) is: \(m = \frac{{{y_2}\; - \;{y_1}}}{{{x_2} - {x_1}}}\)

CALCULATION:

Given: The slope of a line passing through the points (2, 5) and (x, 3) is 2

Here, we have to find the value of x.

As we know that, the slope of the line joining the points (x1, y1) and (x2, y2) is: \(m = \frac{{{y_2}\; - \;{y_1}}}{{{x_2} - {x_1}}}\)

Here, x1 = 2, y1 = 5, x2 = x, y2 = 3 and m = 2.

⇒ \(2 = \frac{{{3}\; - \;{5}}}{{{x} - {2}}}\)

⇒ 2x - 4 = - 2

⇒ 2x = 2

⇒ x = 1

Hence, option B is the correct answer.

CONCEPT:

The slope of a line is given by tan α and it is denoted by m.

i.e m = tan α, where α ≠ π/2 and it represents the angle which a given line makes with respect to the X – axis in the positive direction.

Note:

• Slope of X –axis or a line parallel to X – axis is : m = tan 0° = 0

• Slope of Y – axis or a line parallel to Y – axis is: m = tan π/2 = ∞

• The slope of the line joining the points (x1, y1) and (x2, y2) is: \(m = \frac{{{y_2}\; - \;{y_1}}}{{{x_2} - {x_1}}}\)

• If two lines are parallel then their slope is same.

• If two lines L₁ and L₂ with slopes m₁ and m₂ respectively are perpendicular then m₁ × m₂ = - 1

CALCULATION :

Here, we have to find the value of x such that the lines through the points (- 2, 6) and (x, 8) is perpendicular to the line through the points (3, - 3) and (5, - 9).

Let's find out the slope of the line through the points (3, - 3) and (5, - 9).

As we know that, the slope of the line joining the points (x1, y1) and (x2, y2) is: \(m = \frac{{{y_2}\; - \;{y_1}}}{{{x_2} - {x_1}}}\)

Here, x1 = 3, y1 = - 3, x2 = 5, y2 = - 9.

⇒ \(m_1 = \frac{{{-9}\; - \;{(-3)}}}{{{5} - {3}}} = - 3\)

∵ The lines through the points (- 2, 6) and (x, 8) is perpendicular to the line through the points (3, - 3) and (5, - 9)

So, let the slope of line through the points (- 2, 6) and (x, 8) be m₂

⇒ - 3 × m2 = - 1

⇒ m₂ = 1/3

So, the slope of the line through the points (- 2, 6) and (x, 8) is 1/3

⇒ \(\frac{1}{3} = \frac{{{8}\; - \;{6}}}{{{x} - {(-2)}}}\)

⇒ x + 2 = 6

⇒ x = 4

Hence, option B is the correct answer.

Concept:

Collinearity of three points:

If A, B and C are any three points in the XY-Plane, then they will lie on a line, i.e., collinear if and only if slope of AB = slope of BC

The slope of the line passing through (x1, y1) and (x2, y2) is given by m = \(\rm \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\)

Calculation:

Let four points are  A(-1, -2), O(0, 0), B(1, 2) and C(2, 4)

Now,

m₁ = slope of OA = 2

m₂ = slope of OB = 2

m₃ = slope of OC = 2

Since m1 = m2 = m3 

So, the points (-1, -2), (0, 0), (1, 2) and (2, 4) are collinear

CONCEPT:

The slope of a line is given by tan α and it is denoted by m.

i.e m = tan α, where α ≠ π/2 and it represents the angle which a given line makes with respect to the X – axis in the positive direction.

Note:

• Slope of X –axis or a line parallel to X – axis is : m = tan 0° = 0

• Slope of Y – axis or a line parallel to Y – axis is: m = tan π/2 = ∞

• The slope of the line joining the points (x₁, y₁) and (x₂, y₂) is: \(m = \frac{{{y_2}\; - \;{y_1}}}{{{x_2} - {x_1}}}\)

CALCULATION:

Here, we have to find the slope of the a line which passes through the points (- 2, 3) and (8, - 5)

As we know that, the slope of the line joining the points (x1, y1) and (x2, y2) is: \(m = \frac{{{y_2}\; - \;{y_1}}}{{{x_2} - {x_1}}}\)

Here, x₁ = - 2, y₁ = 3, x₂ = 8 and y₂ = - 5

So, the slope of the given line is \(m = \frac{{{-5}\; - \;{3}}}{{{8} - {(-2)}}} = - \frac{4}{5}\)

Hence, option A is the correct answer.

CONCEPT:

The slope of a line is given by tan α and it is denoted by m.

i.e m = tan α, where α ≠ π/2 and it represents the angle which a given line makes with respect to the X – axis in the positive direction.

Note:

• Slope of X –axis or a line parallel to X – axis is : m = tan 0° = 0

• Slope of Y – axis or a line parallel to Y – axis is: m = tan π/2 = ∞

• The slope of the line joining the points (x1, y1) and (x2, y2) is: \(m = \frac{{{y_2}\; - \;{y_1}}}{{{x_2} - {x_1}}}\)

• If two lines are parallel then their slope is same.

CALCULATION:

Here, we have to find the value of y such that the line through the points (5, y) and (2, 3) is parallel to the line through the points (9, - 2) and (6, - 5)

Let's find the slope of the line  through the points (9, - 2) and (6, - 5)

As we know that, the slope of the line joining the points (x1, y1) and (x2, y2) is: \(m = \frac{{{y_2}\; - \;{y_1}}}{{{x_2} - {x_1}}}\)

Here, x1 = 9, y1 = - 2, x2 = 6, y2 = - 5.

⇒ \(m = \frac{{{-5}\; - \;{(-2)}}}{{{6} - {9}}} = 1\)

∵ The line through the points (5, y) and (2, 3) is parallel to the line through the points (9, - 2) and (6, - 5)

So, the slope of the line through the points (5, y) and (2, 3) is also 1.

So, know x1 = 5, y1 = y, x2 = 2, y2 =  3 and m = 1.

⇒ \(1 = \frac{{{3}\; - \;{y}}}{{{2} - {5}}}\)

⇒ - 3 = 3 - y

⇒ y = 6

Hence, option A is the correct answer.